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Computer Arithmetic of Geometrical Figures

[Category : - Computers and computer accessories ]
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Overview:
This project describes various versions of processors, designed for affine transformations of many-dimensional figures - planar and spatial. This processors is oriented to affine transformation of unstructured geometrical figures with arbitrary points distribution. The type of data presentation used in this project is non-conventional, based on a not well-known theory of vectors and geometrical figures coding.
The original theory, algorithms and units are submitted as the book. The book covers the figures coding theory - the codes structure, algorithms of coding and decoding for planar and spatial figures, arithmetical operations with planar and spatial figures. The theory is supplemented by numerous examples. The arrangement of several versions of geometrical processor is considered - data representation, operating blocks, hardware realization of coding, decoding and arithmetic operations algorithms. The processors internal performance is appraised.

The project includes
-Theory of coding,
-Operations algorithms,
-Examples of coding, decoding and computations,
-Description of several versions of processors,
-A system of commands for them,
-Schemes of operational units,
-Comparative analysis.

Features:
The method of representation of geometrical figures is offered. By this method the set of binary codes of complex numbers and of the vectors is represented by a single binary code. Its volume is considerably smaller that the total volume of the initial binary codes array. The comparative volume reduction depends on the amount of numbers being coded and increases as this amount grows. The coded set of complex numbers is NOT structured. The coded complex numbers and vectors are a coordinate set that the calculations are to be performed with. Any additional information about the points (for instance, their color) is not subject to coding, and should be saved in a separate array - an attribute array. Geometrical code saves (in addition to the coordinates) also the information about every points connection with its attributes.

Benefits:
For comparison we shall consider three types of arithmetic units:
-Traditional, operating with the proposed vector codes and containing several calculators, working simultaneously.
-Vectorial, operating with the proposed vector codes and also containing several calculators, working simultaneously.
-Geometrical, operating with geometrical codes of figures.

Contents :
1. Introduction 8
2. Prototypes 13
2.1. Data Representation 13
2.2. The Simplest Arithmetic Unit 14
2.3. Arithmetic Unit with Rectangular Codes 17
3. Foundations of Computer Arithmetic for Complex Numbers and Vectors 19
3.1. Coding Method for Complex Numbers 19
3.2. Special Algebra in Vector Space 21
3.2.1. Algebra in 3-dimensional vector space
3.2.2. Component-wise multiplication
3.2.3. Vector product
3.2.4. Scalar product
3.2.5. The turning of a vector
3.2.6. Centroaffine transformation
3.2.7. Many-dimensional space
3.3. Two Methods of Multidimensional Vectors Coding 25
3.3.1. Method 1
3.3.2. Method 2
3.4. Algebraic addition of M-codes 29
3.4.1. Multidigit circuits for M-codes
3.4.2. M-code Inverter
3.4.3. M-codes Inverse Adder
3.4.4. M-code Adder
3.4.5. M-code Subtractor
3.4.6. Sign Determinant M-code
3.5. Multiplication of Many-dimensional Vectors 36
3.5.1. Multiplication Method of Many-dimensional Vectors
3.5.2. Multiplication by Base Function to the Radix (3.3.10)
3.5.3. Multiplication by Base Function to the Radix (3.3.7)
3.5.4. Multiplication of the Whole Codes of Vectors to the Radix (3.3.10)
3.5.5. Multiplication of the Whole Codes of Vectors to the Radix (3.3.7)
3.5.6. Componentwise Multiplication of Many-dimensional Vectors
3.6. Scalar and Vector Multiplication 41
3.6.1. Scalar Product
3.6.2. Vector Product
3.6.3. Carries in Scalar Multiplication
3.6.4. Carries in Vector Multiplication
3.7. Algoritms and Devices for CodingDecoding of Complex Numbers and Vectors 48
3.7.1. Coding of Complex Number in System 1
3.7.2. Decoding of Complex Number in System 1
3.7.3. Coding of Complex Number in System 2
3.7.4. Decoding of Complex Number in System 2
3.7.5. Coder of Positive M-code into P-code
3.7.6. Decoder of M-code into P-code
3.7.7. Full Decoder of M-code into P-code
3.7.8. Precoder of P-code into M-code
3.7.9. Partitioning Unit for Parts of the Code
4. Vector Processor 57
4.1. Data Representation and Vector Arithmetic Unit 57
4.2. Comparisons 61
5. Figure Coding theory 65
5.1. Primary Geometrical Codes 68
5.1.1. Data Representation
5.1.2. Arithmetic operations with geometrical codes in a real radix
5.1.2.1. Introduction
5.1.2.2. Writing of Base Code
5.1.2.3. Transpositions
5.1.2.4. Addition of Geometrical and Basic Codes when r=2
5.1.2.5. Algebraic Addition of Geometrical and Basic Codes when r=2
5.1.2.6. Algebraic Addition of Geometrical and Basic Codes when r=-2
5.1.2.7. Multiplication of Geometrical and Basic Codes
5.1.2.8. Division of Geometrical Code by Basic Code
5.1.2.9. Rounding-off of Geometrical Code
5.1.3. Geometrical codes in a complex radix
5.1.3.1. Algebraic Addition of Geometrical and Basic Codes
5.1.3.2. Multiplication of Geometrical and Basic Codes
5.1.4. Coding and transformation of planar figures
5.1.4.1. Method of coding
5.1.4.2. Carry 95
5.1.4.3. Centroaffine transformation
5.1.4.4. Affine transformation
5.1.5. Coding and Transformation of Spatial Figure
5.2. Attribute Geometrical Codes 106
5.2.1. Data Representation
5.2.2. AGC in a real radix
5.2.2.1. Writing of a given Number
5.2.2.2. Writing of a given Value
5.2.2.3. Reading the value of the path with the given number
5.2.2.4. Addition of AGC to the basic code when r=2
5.2.2.5. Inverse addition of AGC to the basic code when r= -2
5.2.2.6. Inversion of AGC when r= -2
5.2.2.7. Algebraic addition of AGC
5.2.2.8. Search for the Next Open Path, its Number and it's Value
5.2.2.9. Multiplication of AGC by the basic code
5.2.3. Attribute geometrical codes in a complex radix
5.2.3.1. Inverse addition with the basic code
5.2.3.2. Invertion
5.2.3.3. Deformation
5.2.4. Attribute Geometrical Codes of Spatial Figures
5.2.5. Contracted attribute geometrical codes
6. Geometrical Processor 115
6.0. Data Presentation 115
6.1. Full Specific Random-access Memory 118
6.2. Fragmentary Specific Random-access Memory 119
6.3. Maximal Arithmetic Unit of Geometrical Figures 123
6.4. Fragmentary Arithmetic Unit of Geometrical Figures 124
6.5. Processor with a Maximal Arithmetical unit 126
6.6. Processor with Fragmentary Arithmetic Unit 129
6.7. The Main Procedures 132
6.7.1. Affine Transformation
6.7.2. Rounding
6.7.3. Rough rounding
6.7.4. Attributes Correction
6.7.5. Attributes Calculation
6.7.6. Coding a Figure
6.7.7. Decoding a Figure
6.8. Operational units 136
6.8.1. Writing unit for the number with the given code
6.8.2. Writing unit for the value with the given code
6.8.3. Reading unit for path value with the given number
6.8.4. Inverse adder
6.8.5. Search unit for the first open path, its numbers and its values
6.8.6. Reading unit for the number and value of the path with the given terminal vertex
6.8.7. Next terminal vertex search unit
7. Comparative Analysis 143
References 149
Designation 150
List of Examples 152
List of Tables 153
List of Figures 155

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solik (Israel)

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